Due to the non-homogeneous and nonlinear nature of the fractional \((p,q)\) -operators, the corresponding nonlocal problems have attracted attention of many researchers in recent years. For instance, see [1]}, [2]}, [3]} for the existence and multiplicity results for problems involving the fractional \((p,q)\) -Laplacian, and we refer [4]} for a survey of recent developments on nonstandard growth problems. On the other hand there is not much literature available regarding the regularity results. Particularly, in [3]}, Goel et al. have obtained the \(C^{0,\alpha }_{\rm loc}\)  regularity result, with some unspecified \(\alpha \in (0,1)\) , for weak bounded solution in the superquadratic case. Further, we mention the recent work of [6]}, where authors proved the global Hölder continuity results (in the spirit of [7]}) for weak solutions to problems involving the fractional \((-\Delta )_g^s\) -Laplacian, where \(g\)  is a convex Young's function. However, this does not include our problem, even for the case \(s_1=s_2\) , because of the power type of growth condition (2.2) and (2.4), there. In [8]}, authors have obtained interior Hölder regularity of viscosity solutions to a class of fractional double phase problems. Recently, in [9]}, authors of the present work have obtained almost optimal global Hölder continuity results for weak solutions to fractional \((p,q)\) -problems in the superquadratic case (\(p\ge q\ge 2\) ). To establish the optimal interior regularity result for local weak solutions (without the boundedness assumption), authors proved the local boundedness result, which uses a new Caccioppoli type inequality for non-homogeneous operators. Then, employing Moser's iteration technique on the discrete differential of the solution and exploiting the local boundedness of the function \(f\) , they proved suitable Besov space inclusion (which in turn gives Hölder continuity). By establishing the asymptotic behavior of the fractional \(q\) -Laplacian (\((-\Delta )_q^{s_2}\) ) of the distance function \(d^{s_1}\)  near the boundary, authors have further obtained almost optimal \(d^{s_1}\)  boundary behavior of the weak solution. This coupled with the interior Hölder regularity result proves the almost optimal \(s_1\) -Hölder continuity result globally in \(\mathbb {R}^N\) .
